Probability theory and stochastic processes Definition

Conditional probability distribution

Builds on Conditional probability Joint probability distribution Marginal distribution Probability mass function Probability density function

Definition

Let $X$ and $Y$ be random variables on one probability space. Discrete case: let $X$ and $Y$ be discrete and let $x$ satisfy $P(X = x) > 0$ . The conditional distribution of $Y$ given $X = x$ has the probability mass function $p_{Y \mid X}(y \mid x) = P(Y = y \mid X = x) = \frac{P(X = x, \, Y = y)}{P(X = x)}.$ Continuous case: let $(X, Y)$ have a joint density $f_{X, Y}$ , and let $f_X$ be the marginal density of $X$ . For each $x$ with $f_X(x) > 0$ , the conditional distribution of $Y$ given $X = x$ has the density $f_{Y \mid X}(y \mid x) = \frac{f_{X, Y}(x, y)}{f_X(x)}.$

Remarks

In the discrete case, $p_{Y \mid X}(\cdot \mid x)$ is the law of $Y$ under the conditional probability $P(\cdot \mid X = x)$ . In both cases, the joint law is the marginal times the conditional: $f_{X, Y}(x, y) = f_X(x) \, f_{Y \mid X}(y \mid x)$ . If $X$ and $Y$ are independent, the conditional distribution of $Y$ given $X = x$ is the law of $Y$ , for each such $x$ . In general measure theory, one defines it as a regular conditional distribution. This is a probability kernel $\kappa$ with $P(X \in A, Y \in B) = \int_A \kappa(x, B) \, \mu_X(dx)$ .

Reading path

To understand this page from the ground up, read these 51 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Random variable with a density definition
Joint probability distribution definition
Probability mass function definition
Probability density function definition
Marginal distribution definition
Conditional probability distribution definition

Tag	`0YVDC5PYDW`
URI	`m2://0YVDC5PYDW`
Kind	definition
Writing	W0 stub
MSC	60E05
Defines	Conditional probability distribution
Uses	Conditional probability, Joint probability distribution, Marginal distribution, Probability mass function, Probability density function

Conditional probability distribution

Definition

Remarks

Reading path

Further reading

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